Algorithms for the construction of incoherent frames under various design constraints

Abstract

Unit norm finite frames are generalizations of orthonormal bases with many applications in signal processing. An important property of a frame is its coherence, a measure of how close any two vectors of the frame are to each other. Low coherence frames are useful in compressed sensing applications. When used as measurement matrices, they successfully recover highly sparse solutions to linear inverse problems. This paper describes algorithms for the design of various low coherence frame types: real, complex, unital (constant magnitude) complex, sparse real and complex, nonnegative real and complex, and harmonic (selection of rows from Fourier matrices). The proposed methods are based on solving a sequence of convex optimization problems that update each vector of the frame. This update reduces the coherence with the other frame vectors, while other constraints on its entries are also imposed. Numerical experiments show the effectiveness of the methods compared to the Welch bound, as well as other competing algorithms, in compressed sensing applications.

Hybrid MIMO Architectures for Millimeter Wave Communications: Phase Shifters or Switches?

“…Alternatively, combinatorial sequences based on almost difference sets [87], can be found for a more general range of dimensions providing near optimal coherence. In any case, numerical algorithms to create highly incoherent frames can be applied [88] [89]. The same designs can be used for the more general hybrid architecture A4 based on switches.…”

A Frame-Theoretic Scheme for Robust Millimeter Wave Channel Estimation

“…This scheme, referred to as sequential iterative decorrelation via convex optimization (SIDCO), was originally introduced in [20] only for frames in R M . More recently, a strategy to generalize SIDCO to frames in C M , referred to as complex SIDCO (C-SIDCO), was reported by [26]. An explicit and complete mathematical formulation of C-SIDCO was, however, not given in [26].…”

“…More recently, a strategy to generalize SIDCO to frames in C M , referred to as complex SIDCO (C-SIDCO), was reported by [26]. An explicit and complete mathematical formulation of C-SIDCO was, however, not given in [26]. A variation of the latter based on an explicit quadratic program is offered below.…”

“…Consider an existent unit-norm frameF ∈ C M ×N . The strategy of C-SIDCO [26] is to minimize mutual coherence by iteratively solving the problem…”

“…For all the above, the explicit quadratic reformulation of C-SIDCO offered above is original, and can be directly coded on top of optimized standard quadratic solvers, unlike the formulation presented in [26].…”